Sr Examen
¿Cómo vas a descomponer esta tan(x)/(2*tan(x)^2+2)+x/2 expresión en fracciones?
Solución
Ha introducido
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tan(x) x
------------- + -
2 2
2*tan (x) + 2
$$\frac{x}{2} + \frac{\tan{\left(x \right)}}{2 \tan^{2}{\left(x \right)} + 2}$$
tan(x)/(2*tan(x)^2 + 2) + x/2
Simplificación general
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x sin(2*x) - + -------- 2 4
$$\frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4}$$
x/2 + sin(2*x)/4
Combinatoria
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2
x + x*tan (x) + tan(x)
----------------------
/ 2 \
2*\1 + tan (x)/
$$\frac{x \tan^{2}{\left(x \right)} + x + \tan{\left(x \right)}}{2 \left(\tan^{2}{\left(x \right)} + 1\right)}$$
(x + x*tan(x)^2 + tan(x))/(2*(1 + tan(x)^2))
Unión de expresiones racionales
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/ 2 \
x*\1 + tan (x)/ + tan(x)
------------------------
/ 2 \
2*\1 + tan (x)/
$$\frac{x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}}{2 \left(\tan^{2}{\left(x \right)} + 1\right)}$$
(x*(1 + tan(x)^2) + tan(x))/(2*(1 + tan(x)^2))
Potencias
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/ I*x -I*x\
x I*\- e + e /
- + ----------------------------------------
2 / 2\
| / I*x -I*x\ |
| 2*\- e + e / | / I*x -I*x\
|2 - -------------------|*\e + e /
| 2 |
| / I*x -I*x\ |
\ \e + e / /
$$\frac{x}{2} + \frac{i \left(- e^{i x} + e^{- i x}\right)}{\left(- \frac{2 \left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 2\right) \left(e^{i x} + e^{- i x}\right)}$$
x/2 + i*(-exp(i*x) + exp(-i*x))/((2 - 2*(-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)*(exp(i*x) + exp(-i*x)))
Parte trigonométrica
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x sin(x)
- + ----------------------
2 / 2 \
| 2*sin (x)|
|2 + ---------|*cos(x)
| 2 |
\ cos (x) /
$$\frac{x}{2} + \frac{\sin{\left(x \right)}}{\left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2\right) \cos{\left(x \right)}}$$
x sec(x)
- + ------------------------------
2 / 2 \
| 2*sec (x) | / pi\
|2 + ------------|*sec|x - --|
| 2/ pi\| \ 2 /
| sec |x - --||
\ \ 2 //
$$\frac{x}{2} + \frac{\sec{\left(x \right)}}{\left(\frac{2 \sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 2\right) \sec{\left(x - \frac{\pi}{2} \right)}}$$
x sin(2*x) - + -------- 2 4
$$\frac{x}{2} + \frac{\sin{\left(2 x \right)}}{4}$$
x sec(x)
- + ----------------------
2 / 2 \
| 2*sec (x)|
|2 + ---------|*csc(x)
| 2 |
\ csc (x) /
$$\frac{x}{2} + \frac{\sec{\left(x \right)}}{\left(2 + \frac{2 \sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(x \right)}}$$
/pi \
csc|-- - x|
x \2 /
- + ---------------------------
2 / 2/pi \\
| 2*csc |-- - x||
| \2 /|
|2 + --------------|*csc(x)
| 2 |
\ csc (x) /
$$\frac{x}{2} + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\left(2 + \frac{2 \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(x \right)}}$$
x tan(x)
- + ---------------
2 / 2 \
2*\1 + tan (x)/
$$\frac{x}{2} + \frac{\tan{\left(x \right)}}{2 \left(\tan^{2}{\left(x \right)} + 1\right)}$$
x 1 - + ---------- 2 4*csc(2*x)
$$\frac{x}{2} + \frac{1}{4 \csc{\left(2 x \right)}}$$
/ pi\
cos|x - --|
x \ 2 /
- + ---------------------------
2 / 2/ pi\\
| 2*cos |x - --||
| \ 2 /|
|2 + --------------|*cos(x)
| 2 |
\ cos (x) /
$$\frac{x}{2} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\left(2 + \frac{2 \cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}$$
2
x 2*sin (x)
- + ------------------------
2 / 4 \
| 8*sin (x)|
|2 + ---------|*sin(2*x)
| 2 |
\ sin (2*x)/
$$\frac{x}{2} + \frac{2 \sin^{2}{\left(x \right)}}{\left(\frac{8 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 2\right) \sin{\left(2 x \right)}}$$
x 1
- + ---------------
2 / pi\
4*sec|2*x - --|
\ 2 /
$$\frac{x}{2} + \frac{1}{4 \sec{\left(2 x - \frac{\pi}{2} \right)}}$$
x cot(x)
- + ---------------
2 / 2 \
2*\1 + cot (x)/
$$\frac{x}{2} + \frac{\cot{\left(x \right)}}{2 \left(\cot^{2}{\left(x \right)} + 1\right)}$$
x 1
- + --------------------
2 / 2 \
|2 + -------|*cot(x)
| 2 |
\ cot (x)/
$$\frac{x}{2} + \frac{1}{\left(2 + \frac{2}{\cot^{2}{\left(x \right)}}\right) \cot{\left(x \right)}}$$
/ pi\
cos|2*x - --|
x \ 2 /
- + -------------
2 4
$$\frac{x}{2} + \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{4}$$
x/2 + cos(2*x - pi/2)/4
Denominador racional
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/ 2 \
2*tan(x) + x*\2 + 2*tan (x)/
----------------------------
2
4 + 4*tan (x)
$$\frac{x \left(2 \tan^{2}{\left(x \right)} + 2\right) + 2 \tan{\left(x \right)}}{4 \tan^{2}{\left(x \right)} + 4}$$
(2*tan(x) + x*(2 + 2*tan(x)^2))/(4 + 4*tan(x)^2)